For any set \(A\) , let \(\left|A \right|\)  be its cardinality. We denote by \(\operatorname{Coag}(\cdot ,\cdot )\) 
the usual coagulation operator on \({P}_{n}^2\)  (Definition 4.2 in [1]}); i.e.
if \(\pi ^{\prime }\in {P}_{m}\)  and \(\pi \in {P}_{n}\)  are such that
\(\left|\pi  \right| \le m\le \infty \)  we call the pair \((\pi ,\pi ^{\prime })\)  an admissible pair, and define the coagulation
of \(\pi \)  by \(\pi ^{\prime }\)  as
\(\operatorname{Coag}(\pi ,\pi ^{\prime })=(\hat{\pi }_1,\hat{\pi }_2,\cdots ),\) 
